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Lesson Presentation
HoltMcDougal
GeometryGeometry
Holt
4-Ext
Proving Constructions Valid
Objective
Use congruent triangles to prove
constructions valid.
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
When performing a compass and straight
edge construction, the compass setting
remains the same width until you change it.
This fact allows you to construct a segment
congruent to a given segment. You can
assume that two distances constructed with
the same compass setting are congruent.
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
The steps in the construction of a figure can
be justified by combining the assumptions of
compass and straightedge constructions and
the postulates and theorems that are used
for proving triangles congruent.
You have learned that there exists exactly
one midpoint on any line segment.
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Remember!
To construct a midpoint, see the
construction of a perpendicular
bisector on p. 172.
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Example 1: Proving the Construction of a Midpoint
Given: Diagram showing the steps in the
construction
Prove: CD  AB
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Example 1 Continued
Statements
Reasons
1. Draw AC, BC.
1. Through any two points
there is exactly one line.
2. AC  BC
2. Same compass setting used
3. AD  BD
3. Same compass setting used
4. CD  CD
4. Reflex. Prop. of 
5. ∆ADC  ∆BDC
5. SSS Steps 2, 3, 4
6. ADC  BDC
6. CPCTC
7. ADC and BDC are rt. s
7.  s that form a lin. pair are
rt. s.
8. CD  AB
8. Def. of 
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Check It Out! Example 1
Given:
Prove: CD is the perpendicular bisector of AB.
Holt McDougal Geometry
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Proving Constructions Valid
Check It Out! Example 1 Continued
Statements
Reasons
1. Draw AC, BC, AD, and BD.
1. Through any two points
there is exactly one line.
2. AC  BC  AD  BD
2. Same compass setting used
3. CD  CD
3. Reflex. Prop. of 
4. ∆ADC  ∆BDC
4. SSS Steps 2, 3
5. ADC  BDC
5. CPCTC
6. CM  CM
6. Reflex. Prop. of 
7. ∆ACM and ∆BCM
7. SAS Steps 2, 5, 6
8. AMC  BMC
8. CPCTC
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Check It Out! Example 1 Continued
Statements
Reasons
9. AMC and BMC are rt. s
9.  s supp.  rt. s
10. AC  BC
10. Def. of 
11. AM  BM
11. CPCTC
12. CD bisects AB
12. Def. of bisector
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Example 2: Proving the Construction of an Angle
Given: diagram showing the steps in the construction
Prove: D  A
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Example 2 Continued
Since there is a straight line through any two
points, you can draw BC and EF. The same
compass setting was used to construct AC,
AB, DF, and DE, so AC  AB  DF  DE.
The same compass setting was used to
construct BC and EF, so BC  EF. Therefore
ABC  DEF by SSS, and D  A by CPCTC.
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Check It Out! Example 2
Prove the construction for bisecting an angle.
Draw BD and CD (through
any two points. there is
exactly one line). Since the
same compass setting was
used, AB  AC and BD  CD.
AD  AD by the Reflexive
Property of Congruence.
So ABD  ACD by SSS, and BAD  CAD by
CPCTC. Therefore AD bisects BAC by the definition
of an angle bisector.
Holt McDougal Geometry
4-Ext
Proving Constructions Valid
Remember!
To review the construction of an
angle congruent to another angle,
see p. 22.
Holt McDougal Geometry